Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $a = \dfrac{2t + 14}{4t^2 - 8t - 252} \times \dfrac{-3t^2 - 21t - 30}{t + 5} $
Answer: First factor out any common factors. $a = \dfrac{2(t + 7)}{4(t^2 - 2t - 63)} \times \dfrac{-3(t^2 + 7t + 10)}{t + 5} $ Then factor the quadratic expressions. $a = \dfrac {2(t + 7)} {4(t + 7)(t - 9)} \times \dfrac {-3(t + 5)(t + 2)} {t + 5} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac {2(t + 7) \times -3(t + 5)(t + 2) } { 4(t + 7)(t - 9) \times (t + 5)} $ $a = \dfrac {-6(t + 5)(t + 2)(t + 7)} {4(t + 7)(t - 9)(t + 5)} $ Notice that $(t + 7)$ and $(t + 5)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac {-6(t + 5)(t + 2)\cancel{(t + 7)}} {4\cancel{(t + 7)}(t - 9)(t + 5)} $ We are dividing by $t + 7$ , so $t + 7 \neq 0$ Therefore, $t \neq -7$ $a = \dfrac {-6\cancel{(t + 5)}(t + 2)\cancel{(t + 7)}} {4\cancel{(t + 7)}(t - 9)\cancel{(t + 5)}} $ We are dividing by $t + 5$ , so $t + 5 \neq 0$ Therefore, $t \neq -5$ $a = \dfrac {-6(t + 2)} {4(t - 9)} $ $ a = \dfrac{-3(t + 2)}{2(t - 9)}; t \neq -7; t \neq -5 $